## Polynomial functions of modular lattices

• A polynomial function $$f : L \to L$$ of a lattice $$\mathcal{L}$$ = $$(L; \land, \lor)$$ is generated by the identity function id $$id(x)=x$$ and the constant functions $$c_a (x) = a$$ (for every $$x \in L$$), $$a \in L$$ by applying the operations $$\land, \lor$$ finitely often. Every polynomial function in one or also in several variables is a monotone function of $$\mathcal{L}$$. If every monotone function of $$\mathcal{L}$$is a polynomial function then $$\mathcal{L}$$ is called orderpolynomially complete. In this paper we give a new characterization of finite order-polynomially lattices. We consider doubly irreducible monotone functions and point out their relation to tolerances, especially to central relations. We introduce chain-compatible lattices and show that they have a non-trivial congruence if they contain a finite interval and an infinite chain. The consequences are two new results. A modular lattice $$\mathcal{L}$$ with a finite interval is order-polynomially complete if and only if $$\mathcal{L}$$ is finite projective geometry. If $$\mathcal{L}$$ is simple modular lattice of infinite length then every nontrivial interval is of infinite length and has the same cardinality as any other nontrivial interval of $$\mathcal{L}$$. In the last sections we show the descriptive power of polynomial functions of lattices and present several applications in geometry.

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Verfasserangaben: Dietmar Schweigert urn:nbn:de:hbz:386-kluedo-50601 Preprints (rote Reihe) des Fachbereich Mathematik (284) Bericht Englisch 10.11.2017 1996 Technische Universität Kaiserslautern 10.11.2017 15 Fachbereich Mathematik 5 Naturwissenschaften und Mathematik / 510 Mathematik Creative Commons 4.0 - Namensnennung, nicht kommerziell, keine Bearbeitung (CC BY-NC-ND 4.0)